Discrete mathematics in continuous life?

Question

Programming

I asked myself the question of the direct use of discrete mathematics.
Let's take the Catalan numbers, yes, they have more than fifty combinatorial interpretations, but do you have an example of a problem where it was really necessary to count, for example, the number of binary trees on N vertices? Have you ever had to determine the number of surjective mappings or solve a recurrence relation in terms of generating functions in some really useful problem?
I would like to see something like problems solved by graph theory: routing, shortest distances. Of course, I understand that the concept of a graph itself and algorithms on it exist only thanks to the concepts of sets and relations, but still, do the above concepts have a vital benefit right here and now? And if there is no such benefit, then I would be happy with any announcements, like: "be patient, around the corner is an elegant use of all the accumulated abstract concepts there and there."
PS I'll pay attention again, I'm not asking about the applicability of discrete mathematics in general, its necessity is obvious. Nor am I asking about the use of graphs (tree structures, traversal algorithms, pathfinding, matching). I askabout using specific sections: recurrence relations , non-elementary combinatorial objects , and any other advanced combinatorial concepts that I haven't gotten to yet that come up in interesting ways in research and development. Describe (!) how and when these things happened in your practice.

Answer 1

Such tasks are more common at the stage of algorithm development. When you need to understand the efficiency of an algorithm, fit a structure into memory, compare different structures or approaches, and choose the best one. It is not always possible to immediately get an estimate of O(N^k), there are more difficult situations. Or such a task as organizing a search with the least number of returns, somehow encoding the found variant ... here the number of mappings can come in handy.
The problem in this question is that when you need to apply some combinatorial concept, then you apply it and move on. This fact is not stored in memory, sometimes in the program too. But if such calculations remain in the program, this is a guaranteed impossibility of supporting the code by anyone else :) Even with comments.

Answer 2

UPD.
While the answer was too long, let me insert an update at the beginning. In general, do you need to be more specific? Okay, re k u rrent relations - any serious algorithms, more precisely - the analysis of their performance. Combinatorics is very often used in its pure form in solving any problems - to count the number of objects, relationships, relationships, and so on. Iterate over them in some order, given that their number is too large to accommodate them all in memory. All calculations in the field of information theory use the apparatus of discrete mathematics and probability theory (probabilistic ensembles, error probability, message alphabet, sets of code words, and so on). Cryptography also likes to use different sections, but mostly simpler ones. Although today, with the growing complexity of cryptography, the complexity of analyzing its algorithms is also growing. This is especially true for post-quantum cryptography. Well, I would like to point out that advanced combinatorial concepts are often used in elementary particle physics to describe their states. And, what's funny, this is the only basic information that can be obtained in quantum mechanics - the rest is just probabilities.
Ready, attention, and... Let's go!
Before trying to condemn mathematicians in the uselessness of their work, it is worth noting that among those same mathematicians, a special Olympiad on the topic "the most useless research in life" has been opened, which not only competes with the Ig Nobel Prize, but often very, very much surpasses it (in terms of the slogan of the Ignobel Prizes themselves - "first they make you laugh, and then you think," although this is not laughter at times, but a banal abuse of the conceptual apparatus).
So let's take physics as an example. Newton lived. In his time, mathematics was incredibly backward, almost all mathematics at that time was the product of Arab scientists, the Arab countries themselves are the birthplace of real mathematics. But besides geometry and accounting, it was rarely used anywhere, and fabrications on the theory of numbers and discontinuous-continuous continuums were not only shallow, but also incomplete. Many things were axiomatic, the evidence base was rather meager... Under such conditions, huge castles were built and trebuchets were constructed, settlements were made in the markets, alchemical studies were carried out, even plots were composed, pictures were drawn, symphonies were played...
Before Newton. He brought the magic of the continuous into mathematics. Before him, mathematics was a collection of rules for operations on numbers. After him, it became a complex device of functions (and what is a function, as a non-unique mapping of the set of arguments of a function to the set of its values, oops, discrete mathematics), sets and continuums. No, not right away. In general, in those days, science was strongly religious, due to which Copernicus paid with his life. But Newton was much more interested in the structure of the world than many others, and theology gave only stupid excuses, but no answers. But mathematics was stingy, and using those means it was impossible to succinctly describe the world around us. Therefore, together with the notorious Leibniz, a system of differential calculus was developed, which perfectly described the world around as the interaction of cumulative forces, including gravity. It is under the influence of gravity, as Newton showed, that we stand on the Earth, the Australians do not fall down, and the Earth itself revolves around the Sun. Moreover, he showed something deeper - the relativity of motion. Subsequently, Einstein will mathematically succinctly describe not only the relativity of motion, but also the relativity in the calculation of time! And also notice the photoelectric effect, which opened the door to quantum mechanics.
Of course, quantum mechanics is very strange in its structure, very discontinuous, sometimes counterintuitive, illogical and... contradictory. Over time, the contradictions gathered and went into the general theory of relativity, which describes the gravitational interaction as a curvature of space-time. Quantum mechanics began to describe strong, weak and electromagnetic forces. It operates with finite, highly discrete sets. Integrals were replaced by sums, smooth graphs - by a finite set of points.
But what has quantum mechanics given us? As quantum mechanics developed, there was an urgent need to perform very complex calculations in a small amount of time. Why? The Second World War. It just so happened that only a couple of nuclear warheads served as a modest gift to that war from quantum (and not so) mechanics, but some fruits were collected a little later. Huge battlecruisers and dreadnoughts with large-caliber weapons and an effective range measured in tens of kilometers. How is this possible? Imagine the atmosphere. It is heterogeneous - with height, the density, and, accordingly, the aerodynamic drag decreases. If you release a volley at an angle of more than 60 degrees (unfortunately, I don’t remember who noticed it, but, emnip, much earlier than World War II), then the range of destruction will increase by more than 4 times. But the targeting of this action leaves much to be desired - it is too difficult to calculate, there are too many variables - humidity, pressure, temperature, not only here, but also at an altitude of ten kilometers, not only here, but also near the target of destruction. The Cold War set the task - to learn how to calculate multi-storey nonlinear equations in less than two or three minutes. The result was several cabinets ... Several dozen cabinets with cathode ray tubes, replacing compact transistors in those days.
But there was something else that haunted the military minds of that time. Message delivery. Everything is good and fine, as long as you can hit the target from a distance of a couple of tens of kilometers, and even for sure. But everything becomes not so rosy if the order to attack can be heard not only by the attacker, but also by the attacked. Once again - a battleship is a vessel a hundred or two meters long. But if it can still be dressed up like a Christmas tree with wires, then it’s worth recalling that this colossus floats for months on the high seas (more precisely, of course, the ocean, but that’s how it is accepted). All this makes you think about the confidentiality of communication. And what kind of brick factories were built during the communication between Churchill and Stalin. After all, who is listening? Yes, even an ally - those conversations were only for two pairs of ears. All the same, the United States took possession of weapons of total destruction.
In general, some methods should also be used here. Comrade Claude Shannon, who formulated the modern foundations of information theory, a branch of mathematics, radio engineering and computer science, seriously took up this issue. Applied mathematics, you can see! He postulated such archival concepts as an absolutely strong cipher, a sufficiently strong cipher, determined the very strength of the cipher, laid the foundation for the search for asymmetric encryption methods (due to which https works). He opened a whole branch of applied science. I am already silent about the noise-immune transmission of information, which very actively exploits entire layers of discrete mathematics and probability theory.
For years, mathematicians wrote useless papers on number theory, believing they were winners of their own Special Olympiad. For years they have been composing n-dimensional spaces. But von Neumann came along and created the computer. Claude Shannon came along and postulated a confidential channel with an arbitrarily small probability of successful transmission of information. Another bunch of theoretical physicists came running and created the M-string theory.
And you say... Continuous life. What if it's very intermittent?

Answer 3

Something suggests that any more or less serious development (whether it be the IT sphere, physics (even more so!), etc.)) requires a serious mathematical apparatus. And there it all unfolds in full. Otherwise, no one would need it, and no one would use it.

Answer 4

First, google what middleware is. How is it written and by whom? When autotests run for hours. When you think for 3 hours, and then you write 15 lines of code and everything works right away, because run and check "like everyone else" stupid no time. Any company that writes its own-foreign-search-other highload-middleware uses all this. Amazon, Google, Facebook, Twitter, Mailru, Yandex, Rumbler, Bing. All-all-all.